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Very often I have a mathematical thought or question, and think to myself "surely someone has thought of this before." However, I'm not really sure how mathematicians search for mathematical work. Some things I've tried:

  • Google search
  • Arxiv search
  • Posting on the internet (here, stack exchange, overflow)
  • ChatGPT (are there any AI tools yet for searching through math literature?)

Sometimes the hard part is not knowing what the standard words are for the the concepts you're thinking of. What is your approach to looking for math? For context: I was thinking about the set of distances between points in a subset of the real line. If we look at some subset A of the reals, and then the set D of all distances between points in A, what sorts of things can we say about D given A? Topology, measure, location, etc. How would I go about finding if theorems of this sort exist?

H_R
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  • You are asking about the Minkowski difference $A-A$. You might start with this and this, for example. Googling that term should get you lots of things ... – Ted Shifrin Nov 20 '24 at 18:56
  • Thank you for your answer, I will check them out. – H_R Nov 20 '24 at 18:58
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    To help whoever attempts to write up an answer, I'd like to ask: You ask how mathematicians search for mathematical work, but you mention "not knowing the standard words" (which I'd think a mathematician in a given field of math would know) and finding "theorems" (which I'd think could often be in a textbook or expository blog post/MathOverflow Answer, rather than having to look through academic papers, which is what I associate with the word "work"). Did you actually intend to ask for how an outsider non-mathematician might find introductory material? That's a very different question. – Mark S. Nov 20 '24 at 19:06
  • For searching specific papers/books and authors (not exactly what you asked, but might be of use to you anyway), this MSE answer might be helpful. – Dave L. Renfro Nov 20 '24 at 20:27
  • Also you can use approach zero website – Guruprasad Nov 20 '24 at 20:38
  • @MarkS. My question is somewhere in between your two questions. Someone with an education in math who knows the standard language, but who is not a researcher. See, e.g., the context I provided. I suppose I am looking for introductory material, if it exists, about a topic that would likely not be covered in a math course at a university. – H_R Nov 21 '24 at 14:24

1 Answers1

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Although there is a huge amount of literature on this topic, I’ll only mention a little known 1939 book that deserves to be better known. Because it seems no one thus far has said much of anything about this book online (3 decades of numerous internet math discussion groups, amazon reviews, etc.), at least as far as I know, I'll try to remedy this with my answer.

Sophie Piccard (1904−1990), Sur les Ensembles de Distances des Ensembles de Points d’un Espace Euclidean [On Sets of Distances for Subsets of Euclidean Space], Mémoires de l Université Neuchâtel #13, Secrétariat de l’Université, Neuchâtel, 1939, 212 pages. MR 2,129d; Zbl 23.01802; JFM 65.1170.03

This book studies distance sets for a pair of subsets $A,\,B$ of ${\mathbb R}^n$ using the notation $D(A;B) = \{x \in \mathbb R:\; \text{there exist} \; a \in A \; \text{and} \; b \in B \; \text{such that} \; x = \|a – b \| \}.$ Chapter 1: Distance sets for isolated, open, closed, scattered, $F_{\sigma},$ $G_{\delta},$ etc. sets are studied with respect to cardinality, Lebesgue measurability, Borel class, etc. Both the well-known result of Steinhaus and Piccard’s Baire category analog (the MR review omits the needed Baire property hypothesis) are proved in Chapter 1. Chapter 2: Results involving the property of a set being congruent with its complement. Chapter 3: Results involving perfect sets, especially in relation to their definition in terms of $n$-ary expansions. Chapter 4: Results involving conditions on a set in order that it will be a distance set for some sets $A$ and $B$ (or a distance set for some set $A=B).$

Reviewed by: Henri Léon Lebesgue (1875−1941), L’Enseignement Mathématique (1) 38 (1939−1940), pp. 361−362 (in French); John Todd (1911−2007), Mathematical Gazette 24 #261 (October 1940), p. 302; Frederick Arthur Ficken (1910−1978), Bulletin of the American Mathematical Society 49 #1 (January 1943), pp. 29−31. The lengthy Mathematical Reivews (MR) review states many of the results proved in this book, and is freely available at the Internet Archive -- 2 #4 (April 1941), pp. 129−130.

Dave L. Renfro
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  • Very nice. Do you know if there is an English translation? – H_R Nov 21 '24 at 14:31
  • @H_R: If such a translation exists (aside from buried in someone’s Masters project or some such), then I'm pretty sure I would have read about it in print somewhere if it was more than 20 years ago, and its existence would surely arise in a google search if done in the last 20 years. However, given its obscurity, I would be VERY surprised if there was an English translation. Indeed, even for something as well known as Cantor's set theory papers from the 1870s and 1880s, only 2 or 3 of over a dozen such papers have been translated to English. – Dave L. Renfro Nov 21 '24 at 16:12